Page:Cantortransfinite.djvu/112

[486] An aggregate with the cardinal number ${\displaystyle {\mathfrak {a}}.{\mathfrak {b}}}$ may also be made up out of two aggregates ${\displaystyle M}$ and ${\displaystyle N}$ with the cardinal numbers ${\displaystyle {\mathfrak {a}}}$ and ${\displaystyle {\mathfrak {b}}}$ according to the following rule: We start from the aggregate ${\displaystyle N}$ and replace in it every element ${\displaystyle n}$ by an aggregate ${\displaystyle M_{n}\sim M}$; if, then, we collect the elements of all these aggregates ${\displaystyle M}$ to a whole ${\displaystyle S}$, we see that

(7)

${\displaystyle S\sim (M.N)}$,

and consequently

For, if, with any given law of correspondence of the two equivalent aggregates ${\displaystyle M}$ and ${\displaystyle M_{n}}$, we denote by ${\displaystyle m}$ the element of ${\displaystyle M}$ which corresponds to the element ${\displaystyle m_{n}}$ of ${\displaystyle M_{n}}$, we have

(8)

${\displaystyle S={m_{n}}}$;

and thus the aggregates ${\displaystyle S}$ and ${\displaystyle (M.N)}$ can be referred reciprocally and univocally to one another by regarding ${\displaystyle m_{n}}$ and ${\displaystyle (m,n)}$ as corresponding elements.

From our definitions result readily the theorems:

(9)

${\displaystyle {\mathfrak {a}}.{\mathfrak {b}}={\mathfrak {b}}.{\mathfrak {a}}}$,

(10)

${\displaystyle {\mathfrak {a}}.({\mathfrak {b}}.{\mathfrak {c}})=({\mathfrak {a}}.{\mathfrak {b}}).{\mathfrak {c}}}$,

(11)

${\displaystyle {\mathfrak {a}}({\mathfrak {b}}+{\mathfrak {c}})={\mathfrak {a}}{\mathfrak {b}}+{\mathfrak {a}}{\mathfrak {c}}}$;

because:

Addition and multiplication of powers are subject,